Question

Languages in Canada Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asia/Pacific region

a. Explain why this is a valid probability model.

b. What is the probability that the chosen person’s mother tongue is not English?

c. What is the probability that the chosen person’s mother tongue is one of Canada’s official languages?

Short answer

Part(a) It is valid probability model as it follows both conditions of probability.

Part(b) $0.37$is the probability that the chosen person’s mother tongue is not English.

Part(c) $0.85$ is the probability that the chosen person’s mother tongue is one of Canada’s official languages.

Step-by-step solution

Part(a) Step 1 : Given information

We are given a table. We need to explain validity of probability model.

Language	English	French	Asian/Pacific	Other
Probability	$0.63$	$0.22$	$0.06$	$0.09$

Part(a) Step 2 : Simplify

As we know sum of all probabilities must be $1$and probabilities should lie between $0$and $1$

Sum of probabilities is $0.63+0.22+0.06+0.09=1$

Now, from the table we can say it follows both above mentioned conditions.

So, We can say the probability model is valid.

Part(b) Step 1 : Given information

We are given a table. We need to find the probability that the chosen person’s mother tongue is not English.

Language	English	French	Asian/Pacific	Other
Probability	$0.63$	$0.22$	$0.06$	$0.09$

Part(b) Step 2 : Simplify

Using complement rule, $\mathrm{P}\left({\mathrm{A}}^{\mathrm{c}}\right)=\mathrm{P}\left(\overline{\mathrm{A}}\right)=1-\mathrm{P}\left(\mathrm{A}\right)$

Probability of English : $\mathrm{P}\left(\mathrm{E}\right)=0.63$

Applying complement rule,

$\mathrm{P}\left({\mathrm{E}}^{\mathrm{c}}\right)=1-\mathrm{P}\left(\mathrm{E}\right)=1-0.63=0.37$

Hence, $0.37$ is the probability that the chosen person’s mother tongue is not English.

Part(c) Step 1 : Given information

We are given a table. We need to find the probability that the chosen person’s mother tongue is one of Canada’s official languages.

Language	English	French	Asian/Pacific	Other
Probability	$0.63$	$0.22$	$0.06$	$0.09$

Part(c) Step 2 : Simplify

Using addition rule of mutually exclusive for disjoint events,

$\mathrm{P}(\mathrm{A}\mathrm{U}\mathrm{B})=\mathrm{P}(\mathrm{A}\mathrm{or}\mathrm{B})=\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)$

Probability of English $\mathrm{P}\left(\mathrm{E}\right)=0.63$

Probability of French $\mathrm{P}\left(\mathrm{F}\right)=0.22$

Now,

$$\begin{gathered} \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=\mathrm{P}\left(\mathrm{E}\right)+\mathrm{P}\left(\mathrm{F}\right) \\ \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=0.63+0.22 \\ \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=0.85 \end{gathered}$$

Hence, $0.85$ is the probability that the chosen person’s mother tongue is one of Canada’s official languages.